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Physical Review C

Reconstituting rotational band structure from deformed boson expansions: Generic treatment

Eugene R. Marshalek

Published May 1, 1987• 10 citations
Publisher

Abstract

The usefulness of perturbative expansions in deformed bosons has been limited by difficulties associated with zero-frequency (Goldstone) modes. A method developed earlier by Marshalek and Weneser to overcome these difficulties was restricted to the case in which the broken-symmetry generators commute. The present paper shows how to extend the method to the non-Abelian case of three-dimensional rotation of axially symmetric deformed systems. Application is made to a generic anharmonic Hamiltonian constructed from random-phase-approximation modes.

DOI
10.1103/physrevc.35.1900

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The usefulness of perturbative expansions in deformed bosons has been limited by difficulties associated with zero-frequency (Goldstone) modes. A method developed earlier by Marshalek and Weneser to overcome these difficulties was restricted to the case in which the broken-symmetry generators commute. The present paper shows how to extend the method to the non-Abelian case of three-dimensional rotation of axially symmetric deformed systems. Application is made to a generic anharmonic Hamiltonian constructed from random-phase-approximation modes.
A method is presented by which a geometrical interpretation can be given for boson models without leaving the domain of quantum mechanics. This approach turns out to be particularly useful in symmetry-preserving treatments of deformed boson systems. A U(3) boson model having an SO(2) symmetry, which is to be preserved throughout, is analyzed in detail. For a well-deformed system, rotational and vibrational modes are identified. A procedure is developed by which the parameters of the geometrical description for the deformed case can be calculated in a series expansion controlled by the inverse of the boson number. The method is compared with the corresponding random phase approximation approach.
As a direct generalization of our previous work, we construct a manifestly rotation invariant Hamiltonian describing a system composed of a tensor boson with spin I = 2 — such as, for example, phonon of nuclear quadrupole surface vibration — in which special cubic and quartic self-interactions are taken into account. All the ground state solutions are obtained. It is shown that these solutions can be classified into three classes: The first is the normal vacuum of boson, which is manifestly rotation symmetry. The second class of solutions can be viewed as generated from a single chosen state by rotating it into an arbitrary direction. The third one is Goldstone modes built on this chosen ground state. This chosen state — an exact ground state solution of our rotation invariant Hamiltonian — is not an eigenstate of angular momentum, but contains infinitely many different angular momentum states in it; thus, typifying “spontaneous breakdown” of “rotation symmetry”. It is further shown that this state can be naturally identified to an “intrinsic” deformed state in nuclear collective model of Bohr and Mottelson. The results are examined from the standpoint of Goldstone theorem and detailed discussions are given to explore how Goldstone theorem is modified in our finite dimensional system.
Modulated symmetries are global symmetries with a spatially dependent unit of charge, such as the dipole symmetry and the exponential symmetry. We give the generic condition for a modulated symmetry to be compatible with translationally symmetric Hamiltonians, which we define as a translationally covariant modulated symmetry (TCMS). For Abelian TCMSs, we prove that their units of charge can only contain multipole, exponential and harmonic components. Particularly, we classify all the one-dimensional TCMSs by real Jordan normal form blocks. We further derive the generic Goldstone action for SSB phases of continuous TCMSs, by which we show that a broken multipole symmetry gives higher-order gapless Goldstone modes, a broken harmonic symmetry gives gapless Goldstone modes at finite momenta, and a broken exponential symmetry gives no gapless Goldstone modes, modifying the conventional Goldstone theorem.
In noncommutative spaces to maintain Bose–Einstein statistics for identical particles at the non-perturbation level described by deformed annihilation-creation operators when the state vector space of identical bosons is constructed by generalizing one-particle quantum mechanics it is explored that the consistent ansatz of commutation relations of phase space variables should simultaneously include space–space non-commutativity and momentum–momentum non-commutativity, and a new type of boson commutation relations at the deformed level is obtained. Consistent perturbation expansions of deformed annihilation-creation operators are obtained. The influence of the new boson commutation relations on dynamics is discussed. The non-perturbation and perturbation property of the orbital angular momentum of two-dimensional system are investigated. Its spectrum possesses fractional eigen values and fractional intervals.
Concerning the energy of harmonic oscillator, a prescription is proposed for making the original form unchanged even after q-deformation. Applicability of the prescription is limited, but, it can be applied to various cases which are well known. 1 typeset using PTPTEX.sty <ver.1.0>An idea of q-deformation serves us an interesting viewpoint for the understanding of dynamics of many-body systems described by boson operators. For the q-deformation, the present authors recently gave a possible scheme in relation to the generalized boson coherent state. 1)-3) It is, in some sense, an extension of the idea given by Penson and Solomon. 4) Before entering the central part of this paper, first, we recapitulate the basic idea of Ref. 1). We treat a boson system described by one kind of boson (ĉ, ĉ ∗). As the deformation of (ĉ, ĉ ∗), the operator (ˆγ, ˆγ ∗ ) is introduced in the definition ˆγ = F ( ˆ N)ĉ, ˆγ ∗ = ĉ ∗ F ( ˆ N). (1) Here, F ( ˆ N) denotes a function of the boson number operator, ˆ N, defined by ˆN = ĉ ∗ ĉ. (2) The function F ( ˆ N) obeys a certain condition which is proper to each case under investigation. The present deformation is characterized by the function F ( ˆ N). The commutation relations [ĉ, ĉ ∗ ] and [ˆγ, ˆγ ∗ ] are as follows: [ĉ, ĉ ∗ ] = 1, [ˆγ, ˆγ ∗ ] = ( ˆ N + 1)F ( ˆ N) 2 − ˆ NF ( ˆ N − 1) 2. (3a) (3b) Later, we discuss the case in which the relation (3b) does not hold. If following our deformed boson scheme, any operator which is expressed in terms of a function (ĉ, ĉ ∗), Ô = O(ĉ, ĉ ∗), is deformed by replacing (ĉ, ĉ ∗ ) with (ˆγ, ˆγ ∗ ): Ôq = O(ˆγ, ˆγ ∗ ). (4) However, in the case of the Hamiltonian, a special consideration is necessary for the replacement
We study an extended Lipkin-Meshkov-Glick model that permits a transition to a deformed phase with a broken continuous symmetry. Unlike simpler models, one sees a persistent zero-frequency Goldstone mode past the transition point into the deformed phase. We found that the RPA formula for the correlation energy provides a useful correction to the Hartree-Fock energy when the number of particle N satisfies N>3,N>3,N>3, and becomes accurate for large N. We conclude that the RPA correlation energy formula offers a promising way to improve the Hartree-Fock energy in a systematic theory of nuclear binding energies.